Question

. Verify that the function f satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.

(a) f(x) = 7x − sin(2x), [− π /2 , π /2 ].

(b) f(x) = (2x−1)/( 3x+2) , [0, 2].

Answer 1

Step 1:

(a)

The function is , .

Mean value theorem :

Let f be a function that satisfies the following three hypotheses :

1. f is continuous on .

2. f is differentiable on .

Then there is a number c in such that  .

Step 2:

The function is .

The function is continuous on the interval .

Differentiate  with respect to .

The function is differentiable on the interval .

Then .

Step 3:

From the mean value theorem :

.

Step 4:

Substitute in .

General solution of is .

If then .

If then .

are not in the interval , hence they are not considered.

Solution:

.

Answer 2

Step 1:

(b)

The function is , .

Mean value theorem :

Let f be a function that satisfies the following three hypotheses :

1. f is continuous on

2. f is differentiable on

Then there is a number c in such that .

Step 2:

The function is .

The function is continuous on the interval .

Differentiate with respect to .

The function is differentiable on the interval .

Then .

Step 3:

From the mean value theorem :

.

Step 4:

Substitute in .

are not in the interval , hence it is not considered.

Solution:

.